Theorem sbthlem1 8070

Description: Lemma for sbth 8080. (Contributed by NM, 22-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	⊢ 𝐴 ∈ V
sbthlem.2	⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}

Assertion

Ref	Expression
sbthlem1	⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔

Allowed substitution hints:   𝐴(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐷(𝑓,𝑔)

Proof of Theorem sbthlem1

Step	Hyp	Ref	Expression
1		unissb 4469	. 2 ⊢ (∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ↔ ∀𝑥 ∈ 𝐷 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
2		sbthlem.2	. . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
3	2	abeq2i 2735	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
4		difss2 3739	. . . . . . 7 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴)
5		ssconb 3743	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴) → (𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ↔ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)))
6	5	exbiri 652	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))))
7	4, 6	syl5 34	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))))
8	7	pm2.43d 53	. . . . 5 ⊢ (𝑥 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))))))
9	8	imp 445	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)) → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))
10	3, 9	sylbi 207	. . 3 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))))
11		elssuni 4467	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ ∪ 𝐷)
12		imass2 5501	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐷 → (𝑓 “ 𝑥) ⊆ (𝑓 “ ∪ 𝐷))
13		sscon 3744	. . . . 5 ⊢ ((𝑓 “ 𝑥) ⊆ (𝑓 “ ∪ 𝐷) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)))
14	11, 12, 13	3syl 18	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)))
15		imass2 5501	. . . 4 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ (𝐵 ∖ (𝑓 “ 𝑥)) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))))
16		sscon 3744	. . . 4 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
17	14, 15, 16	3syl 18	. . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
18	10, 17	sstrd 3613	. 2 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
19	1, 18	mprgbir 2927	1 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∪ cuni 4436   “ cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  sbthlem2  8071  sbthlem3  8072  sbthlem5  8074